Noise requirements will be key design drivers in the development of the new generations of propeller-driven aircraft, helicopters and ships. Therefore aeroacoustics and hydroacoustics become increasingly important scientific branches since accurate acoustic predictions are an essential tool for the required Design for Reduced Noise Generation. Generally speaking, the prediction of aerodynamically and hydrodynamically generated noise can be considered as an aerodynamic/hydrodynamic analysis followed by an acoustic one. The present thesis focuses on the development of acoustic formulations based on the Ffocws Williams and Hawkings equation (FWHE), to describe the structure of the noise field induced by propeller driven aeronautical and naval craft, both in the unbounded space and in the presence of scattering bodies, like a fuselage or hull. The reason why the FWHE is at the basis of the developed acoustic formulations is its proven capabiity of providing physically consistent aeroacoustic predictions. Literature shows that, in the aeronautical context, the FWHE is a very efficient aeroacoustic tool allowing the prediction of the fluctuating pressure field induced by rotors and propellers, both for subsonic and transonic flight conditions. The present thesis focuses on the development of acoustic formulations based on the Ffocws Williams and Hawkings equation (FWHE), to describe the structure of the noise field induced by propeller driven aeronautical and naval craft, both in the unbounded space and in the presence of scattering bodies, like a fuselage or hull. The reason why the FWHE is at the basis of the developed acoustic formulations is its proven capability of providing physically consistent aeroacoustic predictions. Literature shows that, in the aeronautical context, the FWHE is a very efficient aeroacoustic tool allowing the prediction of the fluctuating pressure field induced by rotors and propellers, both for subsonic and transonic flight conditions. Although the modelling of noise generation and propagation in the naval context is as complicated as in aeronautics, most of the hydroacoustics analysis of non-cavitating and cavitating propellers is based on the unsteady Bernoulli equation. For this thesis, therefore, it was decided to first apply the FWHE for the prediction of noise generated by naval propellers in unbounded space. A comparison between the FWH-based and the Bernoulli-based approach has been carried out using potential flow assumptions. A novel formulation based on the porous form of the FWHE has been developed to predict the sound radiated by a cavitating propeller subjected to non-uniform inflow. The comparison has been performed both theoretically and numerically. A non-cavitating naval propeller, subjected to a uniform onset flow, has been analyzed. Observing that typical naval operating conditions are such that non-linear terms may be coherently neglected in both formulations, no hydrodynamic input concerning the flow-field around the propeller is required. The Laplace equation for the velocity potential has been solved through a boundary integral formulation and a zero-order boundary integral method (BEM) has been applied as discretization strategy. Using the velocity potential and pressure field on the propeller surface, numerical hydroacoustics investigations showed that the assumed shape of the potential wake has a large influence on the pressure disturbance evaluated by means of the Bernoulli equation. The results obtained with the FWHE, however, are not affected by the assumed wake because here the wake contributes to the noise field only through its indirect effects on the loading noise term. The introduction of free wake modelling resolves the discrepancies in the hydroacoustics results from a theoretical point of view, but introduces numerical problems because the introduction of a free wake leads to a very low rate of convergence in the evaluation of the velocity field compared to the analysis with a prescribed wake model. Because of the apparent high potential of the FWHE a novel formulation of this FWHE was developed aiming at the evaluation of noise generated by cavitation, especially sheet cavitation. This specific type occurs in real operating conditions with a propeller working in the wake of the hull, and governs the low-frequency range of the spectrum of cavitation noise. In this range, a significant contribution to the far field noise is associated with frequencies proportional to the blade passage frequency (the tonal spectrum). The evaluation of the noise due to the cyclic growth and collapse of the cavity on the surface of the propeller in a non-uniform onset flow has been performed through a coupled approach involving the permeable form of the FWHE and a suitable hydrodynamic model describing the unsteady cavitation pattern. This model, called Transpiration Velocity Model (TVM) simulates the presence and the acoustic behaviour of the bubble through the difference between the normal component of the body velocity and the fluid velocity wherever cavitation occurs. This way of treating the impulsive noise radiation far away from cavitating propellers is consistent with the physics of the phenomenon and does not introduce approximations incompatible with a formulation derived under the assumption of rigid surfaces. Numerical results provided by the TVM compared satisfactorily with those provided by the Equivalent Blade Modeling (EBM) which is also based on the FWHE written for impermeable surfaces and that, nowadays, represents the single application, presented in literature, of the acoustic analogy to cavitation noise. The discrepancies in noise prediction arise from the different sensibility of the two approaches to the hydrodynamic data describing the cavitation pattern. Numerical investigations outline that the TVM is more sensitive to the accuracy of the hydrodynamic input because of the need to com- pute time derivatives of the function describing the cavity thickness distribution on the blade surface. For highly impulsive signals, the computation of time derivatives up to the second order may become a very difficult task. Contrarily, the EBM approach based on a step-by step strategy in computing the acoustic effect associated with the vapour cavity dynamics needs only the knowledge of the time-history of the cavity volume on the blade, but exhibits a limited capability to correctly describe rapidly changing flow conditions. In this context, it is worth noting that both TVM and EBM model have been used here with hydrodynamic input from a surface tracking approach to describe the liquid vapour interface as a regular surface defined over cavitating propeller blades. However, from a general standpoint, the FWHE may be coupled to more general two-phase flow solvers through a different use of the porous formulation. In fact, by coupling the hydrodynamics input on a suitable surface, enclosing the two-phase region, with the FWHE used as a Kirchoff formulation , it is possible to model noise sources located in the flow field and associated with distributed vapour pockets. This fact highlights the generality of the FWHE approach. In the described hydroacoustics investigations dealing with noise radiation from an acoustic source (the propeller) the boundary integral solution of the FWHE has always been used as an integral representation, exploiting the knowledge of the hydrodynamic quantities appearing in the kernel of the thickness noise and loading noise terms. The nature of the integral solution of the FWHE changes when the emphasis is on the scattering effects caused by the presence of bodies in the path of the travelling acoustic waves emitted from the propeller or rotor. In order to appreciate the sound field change when solid surfaces are present in the flow field and to allow the prediction of the noise produced by those aeronautical and naval configurations where one single body may be identified as the main noise source (assuming the pressure on the body independent of the presence of the other bodies), the problem of scattering has been investigated through a novel integral formulation based on the FWHE. A scattering model allows studying the acoustic behaviour of configurations like fuselagepropeller (aircraft), fuselagemain/tailrotor (helicopters) and hullpropeller (ships), without invoking the interactive aerohydrodynamics to calculate the scattered pressure field on the boundary of the scatterer. Differently from noise radiation problems where the FWHE is used as an integral representation, in this problem the integral solution of the FWHE is used as an integral equation to determine the scattered pressure distribution upon the scattering body. The proposed FWH formulation may be applied to those aeronautical or naval multibody configurations where the sources of noise may be considered aerodynamically or hydrodynamically independent on the presence of the rest of the configuration. For some operating conditions, propellerdriven aircraft, rotorcrafts and ships fall in this category. The evaluation of the sound field produced by the impingement of the pressure disturbance(s) on the scatterer(s) requires a prior analysis of the isolated source(s), to identify the incident pressure field(s). The formulation herein proposed is flexible in that it allows to study scattering problems concerning rigid as well as elastic bodies both moving and at rest. Numerical results show that, for stationary rigid or vibrating scattering bodies, the proposed methodol- ogy yields excellent results when simple configurations (for which analytical solutions exist) are investigated. Dealing with moving scatterers, the problem of the quadrupole term must be pointed out because the assumption to ignore the quadrupole term in the FWHE may become too restrictive. Permission to neglect the quadrupole term depends on the advance speed of the scatterer and on its shape. Hence, the analysis of moving scatterers has to be addressed carefully because the Lighthill tensor could give rise to perturbation terms which might become relevant when the integral formulation is used as an integral equation. The importance of the quadrupole contribution in the FWHE must be stressed also for the previous described radiation cases. It should be noted that numerical investigations performed throughout the thesis have been carried out neglecting the quadrupole contribution in the FWHE. The quadrupole contribution is, in principle, important for several reasons. First, it fully describes the acoustic effect of the potential wake. In order to compare the FWHE and the Bernoulli approach exactly, non-linear terms should be included in both formulations. The non-linearities in both methods are not equivalent , that is, some non-linear effects described by the Lighthill tensor in the FWHE are not accounted for by the non-linear terms in the Bernoulli method. Furthermore, the inclusion of the quadrupole term would account for acoustic effects related to cavitating phenomena occurring in the flowfield, like cavitating tip vortices and hub vortices, and bubble cavitation. However, even with neglecting quadrupole terms, numerical results show that the FWHE is an efficient mathematical model for the study of acoustic problems concerning acoustic radiation and scattering for a wide range of applications. A conjecture has been made and motivated that some of the discrepancies between FWHE and other formulations may be justified invoking the presence of the quadrupole term. Hence, for further development and improvement of the present work, a careful investigation of mathematical and computational aspects related to evaluating quadrupole contributions should be considered. In addition, the application of the present methodology to more realistic configurations could require the use of aero/hydrodynamic solvers able to take into account viscousflow effects.
Acoustic Formulations for Aeronautical and Naval Rotorcraft Noise Prediction Based on the Ffowcs Williams and Hawkings Equation
2008
Abstract
Noise requirements will be key design drivers in the development of the new generations of propeller-driven aircraft, helicopters and ships. Therefore aeroacoustics and hydroacoustics become increasingly important scientific branches since accurate acoustic predictions are an essential tool for the required Design for Reduced Noise Generation. Generally speaking, the prediction of aerodynamically and hydrodynamically generated noise can be considered as an aerodynamic/hydrodynamic analysis followed by an acoustic one. The present thesis focuses on the development of acoustic formulations based on the Ffocws Williams and Hawkings equation (FWHE), to describe the structure of the noise field induced by propeller driven aeronautical and naval craft, both in the unbounded space and in the presence of scattering bodies, like a fuselage or hull. The reason why the FWHE is at the basis of the developed acoustic formulations is its proven capabiity of providing physically consistent aeroacoustic predictions. Literature shows that, in the aeronautical context, the FWHE is a very efficient aeroacoustic tool allowing the prediction of the fluctuating pressure field induced by rotors and propellers, both for subsonic and transonic flight conditions. The present thesis focuses on the development of acoustic formulations based on the Ffocws Williams and Hawkings equation (FWHE), to describe the structure of the noise field induced by propeller driven aeronautical and naval craft, both in the unbounded space and in the presence of scattering bodies, like a fuselage or hull. The reason why the FWHE is at the basis of the developed acoustic formulations is its proven capability of providing physically consistent aeroacoustic predictions. Literature shows that, in the aeronautical context, the FWHE is a very efficient aeroacoustic tool allowing the prediction of the fluctuating pressure field induced by rotors and propellers, both for subsonic and transonic flight conditions. Although the modelling of noise generation and propagation in the naval context is as complicated as in aeronautics, most of the hydroacoustics analysis of non-cavitating and cavitating propellers is based on the unsteady Bernoulli equation. For this thesis, therefore, it was decided to first apply the FWHE for the prediction of noise generated by naval propellers in unbounded space. A comparison between the FWH-based and the Bernoulli-based approach has been carried out using potential flow assumptions. A novel formulation based on the porous form of the FWHE has been developed to predict the sound radiated by a cavitating propeller subjected to non-uniform inflow. The comparison has been performed both theoretically and numerically. A non-cavitating naval propeller, subjected to a uniform onset flow, has been analyzed. Observing that typical naval operating conditions are such that non-linear terms may be coherently neglected in both formulations, no hydrodynamic input concerning the flow-field around the propeller is required. The Laplace equation for the velocity potential has been solved through a boundary integral formulation and a zero-order boundary integral method (BEM) has been applied as discretization strategy. Using the velocity potential and pressure field on the propeller surface, numerical hydroacoustics investigations showed that the assumed shape of the potential wake has a large influence on the pressure disturbance evaluated by means of the Bernoulli equation. The results obtained with the FWHE, however, are not affected by the assumed wake because here the wake contributes to the noise field only through its indirect effects on the loading noise term. The introduction of free wake modelling resolves the discrepancies in the hydroacoustics results from a theoretical point of view, but introduces numerical problems because the introduction of a free wake leads to a very low rate of convergence in the evaluation of the velocity field compared to the analysis with a prescribed wake model. Because of the apparent high potential of the FWHE a novel formulation of this FWHE was developed aiming at the evaluation of noise generated by cavitation, especially sheet cavitation. This specific type occurs in real operating conditions with a propeller working in the wake of the hull, and governs the low-frequency range of the spectrum of cavitation noise. In this range, a significant contribution to the far field noise is associated with frequencies proportional to the blade passage frequency (the tonal spectrum). The evaluation of the noise due to the cyclic growth and collapse of the cavity on the surface of the propeller in a non-uniform onset flow has been performed through a coupled approach involving the permeable form of the FWHE and a suitable hydrodynamic model describing the unsteady cavitation pattern. This model, called Transpiration Velocity Model (TVM) simulates the presence and the acoustic behaviour of the bubble through the difference between the normal component of the body velocity and the fluid velocity wherever cavitation occurs. This way of treating the impulsive noise radiation far away from cavitating propellers is consistent with the physics of the phenomenon and does not introduce approximations incompatible with a formulation derived under the assumption of rigid surfaces. Numerical results provided by the TVM compared satisfactorily with those provided by the Equivalent Blade Modeling (EBM) which is also based on the FWHE written for impermeable surfaces and that, nowadays, represents the single application, presented in literature, of the acoustic analogy to cavitation noise. The discrepancies in noise prediction arise from the different sensibility of the two approaches to the hydrodynamic data describing the cavitation pattern. Numerical investigations outline that the TVM is more sensitive to the accuracy of the hydrodynamic input because of the need to com- pute time derivatives of the function describing the cavity thickness distribution on the blade surface. For highly impulsive signals, the computation of time derivatives up to the second order may become a very difficult task. Contrarily, the EBM approach based on a step-by step strategy in computing the acoustic effect associated with the vapour cavity dynamics needs only the knowledge of the time-history of the cavity volume on the blade, but exhibits a limited capability to correctly describe rapidly changing flow conditions. In this context, it is worth noting that both TVM and EBM model have been used here with hydrodynamic input from a surface tracking approach to describe the liquid vapour interface as a regular surface defined over cavitating propeller blades. However, from a general standpoint, the FWHE may be coupled to more general two-phase flow solvers through a different use of the porous formulation. In fact, by coupling the hydrodynamics input on a suitable surface, enclosing the two-phase region, with the FWHE used as a Kirchoff formulation , it is possible to model noise sources located in the flow field and associated with distributed vapour pockets. This fact highlights the generality of the FWHE approach. In the described hydroacoustics investigations dealing with noise radiation from an acoustic source (the propeller) the boundary integral solution of the FWHE has always been used as an integral representation, exploiting the knowledge of the hydrodynamic quantities appearing in the kernel of the thickness noise and loading noise terms. The nature of the integral solution of the FWHE changes when the emphasis is on the scattering effects caused by the presence of bodies in the path of the travelling acoustic waves emitted from the propeller or rotor. In order to appreciate the sound field change when solid surfaces are present in the flow field and to allow the prediction of the noise produced by those aeronautical and naval configurations where one single body may be identified as the main noise source (assuming the pressure on the body independent of the presence of the other bodies), the problem of scattering has been investigated through a novel integral formulation based on the FWHE. A scattering model allows studying the acoustic behaviour of configurations like fuselagepropeller (aircraft), fuselagemain/tailrotor (helicopters) and hullpropeller (ships), without invoking the interactive aerohydrodynamics to calculate the scattered pressure field on the boundary of the scatterer. Differently from noise radiation problems where the FWHE is used as an integral representation, in this problem the integral solution of the FWHE is used as an integral equation to determine the scattered pressure distribution upon the scattering body. The proposed FWH formulation may be applied to those aeronautical or naval multibody configurations where the sources of noise may be considered aerodynamically or hydrodynamically independent on the presence of the rest of the configuration. For some operating conditions, propellerdriven aircraft, rotorcrafts and ships fall in this category. The evaluation of the sound field produced by the impingement of the pressure disturbance(s) on the scatterer(s) requires a prior analysis of the isolated source(s), to identify the incident pressure field(s). The formulation herein proposed is flexible in that it allows to study scattering problems concerning rigid as well as elastic bodies both moving and at rest. Numerical results show that, for stationary rigid or vibrating scattering bodies, the proposed methodol- ogy yields excellent results when simple configurations (for which analytical solutions exist) are investigated. Dealing with moving scatterers, the problem of the quadrupole term must be pointed out because the assumption to ignore the quadrupole term in the FWHE may become too restrictive. Permission to neglect the quadrupole term depends on the advance speed of the scatterer and on its shape. Hence, the analysis of moving scatterers has to be addressed carefully because the Lighthill tensor could give rise to perturbation terms which might become relevant when the integral formulation is used as an integral equation. The importance of the quadrupole contribution in the FWHE must be stressed also for the previous described radiation cases. It should be noted that numerical investigations performed throughout the thesis have been carried out neglecting the quadrupole contribution in the FWHE. The quadrupole contribution is, in principle, important for several reasons. First, it fully describes the acoustic effect of the potential wake. In order to compare the FWHE and the Bernoulli approach exactly, non-linear terms should be included in both formulations. The non-linearities in both methods are not equivalent , that is, some non-linear effects described by the Lighthill tensor in the FWHE are not accounted for by the non-linear terms in the Bernoulli method. Furthermore, the inclusion of the quadrupole term would account for acoustic effects related to cavitating phenomena occurring in the flowfield, like cavitating tip vortices and hub vortices, and bubble cavitation. However, even with neglecting quadrupole terms, numerical results show that the FWHE is an efficient mathematical model for the study of acoustic problems concerning acoustic radiation and scattering for a wide range of applications. A conjecture has been made and motivated that some of the discrepancies between FWHE and other formulations may be justified invoking the presence of the quadrupole term. Hence, for further development and improvement of the present work, a careful investigation of mathematical and computational aspects related to evaluating quadrupole contributions should be considered. In addition, the application of the present methodology to more realistic configurations could require the use of aero/hydrodynamic solvers able to take into account viscousflow effects.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
